The purpose of this paper is to assert the importance of mathematical thinking on orientation of geometrical objects. Our main contents are summarized as follows. H. Freudenthal once pointed out, with various reasons, that orientation of spaces and figures is significant in mathematical thinking, in his study of didactical phenomenology. We assert that the current curriculum has few materials which instruct students the efficiency of the positive use of orientation, although it contains many materials which are related to orientation. We investigate, through several questions to some university students, how they are conscious of mathematical thinking when they face some issues concerning orientation. As a result, we can admit some tendency that many students are unconscious of mathematical methods on orientation. Our next assertion is that mathematical thinking of orientation enables students to recognize polarities, as negative direction for instance. We offer some mathematical problems which are related to some 2 color's property of diagrams and might be difficult to be solved without any idea of orientation. We imposed those problems on the students in the above questions. As a result, we could only find some partial answers, but no complete solutions. It implies that mathematical thinking of orientaion gives students a good opportunity to experience the efficiency of polarities.
